Dual Decision Processes:
Retrieving Preferences when some Choices are Automatic
Francesco Cerigioni∗
Universitat Pompeu Fabra and Barcelona GSE
January 13, 2017
Abstract
Evidence from cognitive sciences suggests that some choices are conscious and reflect
individual volition while others tend to be automatic, driven by analogies with past
experiences. Under these circumstances, usual economic modeling might not be apt
because not all choices are the consequence of individual tastes. We propose a behavioral model that can be used in standard economic analysis that formalizes how
conscious and automatic choices arise by presenting a decision maker composed of two
selves. One self compares past decision problems with the one the decision maker
faces and, when the problems are similar enough, it replicates past behavior (Automatic choices). Otherwise, a second self is activated and preferences are maximized
(Conscious choices). We then present a novel method capable of identifying a set of
conscious choices from observed behavior and finally, we discuss its importance as a
framework to study empirical puzzles in different settings.
(JEL D01, D03, D60)
Keywords:
Dual Processes, Fluency, Similarity, Revealed Preferences, Memory, Automatic Choice
∗
Department of Economics and Business, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, 08005, Barcelona,
Spain. E-mail: [email protected]. The paper has previously circulated under the title “Dual Decision
Processes: Retrieving Preferences when some Choices are Intuitive”. I am grateful to the Ministerio de Ciencia y
Innovacion for the FPI scholarship connected with the project ECO2010-17943. I thank Miguel A. Ballester for his
insightful comments that were essential for the development of this paper and the editor Joel Sobel and four anonymous
referees for their critiques and suggestions that greatly improved this work. Furthermore, I want to acknowledge the
incredible environment I found in The Eitan Berglas School of Economics (TAU). In particular, I am in debt to Kfir
Eliaz, Itzhak Gilboa, Ariel Rubinstein and Rani Spiegler for their help and suggestions. I am also grateful to David
Ahn, Larbi Alaoui, Jose Apesteguia, B. Douglas Bernheim, Caterina Calsamiglia, Andrew Caplin, Laurens Cherchye,
Vincent Crawford, Tugce Cuhadaroglu, Jan Eeckhout, Matthew Ellman, Fabrizio Germano, Johannes Gierlinger,
Benjamin Golub, Edgardo Lara-Cordova, David Laibson, Marco Mariotti, Daniel Martin, Yusufcan Masatlioglu,
Antonio Miralles, Isabel Melguizo, Muriel Nierderle, Efe Ok, Dilan Okcuoglu, Marco Pelliccia, Pedro Rey-Biel, Tomás
Rodrı́guez-Barraquer, Michael Richter, Guillem Roig-Roig, John Tsoukalis and participants at SITE and BRIC for
their comments and suggestions. All mistakes are mine.
1
1
Introduction
Behavioral economics has posed a serious challenge to standard economic theory
since it has documented and studied numerous behaviors inconsistent with preference
maximization. Inconsistencies tend to arise when individuals do not exert the effort
to consciously analyze the problem at hand, e.g., Carroll et al. (2009). A key question
arises then, when are people choosing consciously?
In this paper we provide a first theoretical framework to think about the coexistence of conscious and automatic, i.e., unconscious, behavior and we provide a method
to understand the nature of individual decisions in different situations once this duality is taken into account. Studying this dichotomy is crucial to understand market
outcomes. In fact, as highlighted by Simon (1987) and Kahneman (2002), experts
such as managers, doctors, traders or policy makers often make automatic decisions.
Nevertheless, the sources of automatic choices are still understudied in economics.1
Evidence from cognitive sciences suggests that the familiarity of the choice environment is the key determinant of the relationship between conscious and automatic
choices.2 Unconscious, automatic choices are made in familiar environments. Thus,
if we want to understand the role of automatic decisions in markets, we need (i) to
have a model of automatic choices that takes into account the role of familiarity of
the decision environment and (ii) to analyze whether it is possible from observed
behavior to distinguish between conscious and automatic choices to understand the
preferences that drive observed behavior. These are the research questions we address
in the paper.
To answer the first question we propose a simple formalization of evidence coming
from cognitive sciences whose main contribution is to provide a first model describing
when and how choices should be conscious or automatic. In Section 3, we present a
decision maker described by a simple procedure. Whenever the decision environment
is familiar, i.e., whenever its similarity with what has been experienced, measured by
1
Nonetheless, in economics there is a growing attention to conscious and intuitive reasoning.
See for example Rubinstein (2007) and Rubinstein (2016) for a distinction between conscious and
intuitive strategic choices for players participating in a game or the distinction in Cunningham and
de Quidt (2015) between implicit and explicit attitudes.
2
See section 3.2 for a review of the literature discussing this dichotomy.
2
a similarity function, passes a certain threshold, past behavior is replicated. This is
the source of automatic choices. Otherwise, the best option is chosen by maximizing
a rational preference relation. This is the source of conscious choices. Think for
example of a consumer that buys a bundle of products from the shelves of a supermarket. The first time he faces the shelves, he tries to find the bundle that best fits
his preferences. Afterwards, if the price and arrangement of products do not change
too much, he will perceive the decision environment as familiar and so he will automatically stick to the past chosen bundle. If on the contrary, the change in price
and arrangement of the products is evident to him, he will choose by maximizing his
preferences again.
Even in such a simple framework, there is no trivial way to distinguish which
choices are made automatically and which ones are made consciously. Following the
example, suppose our consumer faces again the same problem but this time a new
bundle is available and he sticks with the old choice. Is it because the old bundle is
preferred to the new one? Or is it because he is choosing automatically?
We show how to find conscious choices and hence restore standard revealed preference analysis by understanding which environments were not familiar. Section 4
assumes that (i) the decision maker behaves according to our model and (ii) the similarity function is known while the threshold is not.3 We then show that, for every
sequence of decision problems, it is possible to identify by means of an algorithm a
set of conscious observations and an interval in which the similarity threshold should
lie. That is, we provide a novel method to restore revealed preference analysis.
First notice that new observations, i.e., those in which the choice is an alternative
that had never been chosen before, must be conscious. No past behavior could have
been replicated. Starting from these observations, the algorithm iterates the following
idea. If an observation is consciously generated, any other less familiar observation,
that is any problem which is less similar to those decision problems that preceded it,
must be also consciously generated. Returning to our consumer, if we know that after
a change in the price of the products on the shelf, the consumer chose consciously,
then he must have done so also in all those periods where the change was even more
3
See Sections 3 and 4 for a justification of the latter hypothesis.
3
evident.
The algorithm identifies a set of automatic decisions in a similar fashion, that
is, first it highlights some decisions that have to be automatic and then finds more
familiar observations to reveal other automatic decisions. Notice that understanding if some decisions were made automatically is very important to understand how
familiarity of environments is determined. Even if automatic choices do not reveal
individual preferences, they tell us what problems are considered familiar, i.e., similar
enough, by the decision maker hence allowing for the identification of the interval in
which the similarity threshold should lie.
The algorithm assumes that the decision maker behaves following our model,
hence falsifiability of the model becomes a central concern. In Section 5 we propose a
testable condition that is a weakening of the Strong Axiom of Revealed Preference that
characterizes our model and thus renders it falsifiable. Section 6 studies the model
as a general framework formalizing the coexistence of behavioral inertia and adaptive
behavior that might give new insights into physicians behavior, pricing behavior of
retailers and portfolio decisions of traders in financial markets. Section 7 concludes
while the Appendix contains an extension of the identification algorithm assuming
only partial knowledge of similarity comparisons.
2
Related Literature
In our model the presence of similarity comparisons makes behavior automatic, that
is, if two environments are similar enough then behavior is replicated. This is a
different approach with respect to the theory for decisions under uncertainty proposed
in Gilboa and Schmeidler (1995) and summarized in Gilboa and Schmeidler (2001).
In case-based decision theory, as in our model, a decision maker uses a similarity
function in order to assess how much alike are the problem he is facing and the ones
he has in his memory. In that model the decision maker tends to choose the action
that performed better in past similar cases. There are two main differences with the
approach we propose here. First, from a conceptual standpoint, our model relies on
the idea of two selves interacting during the decision making process. Second, from a
4
technical point of view, our model uses the similarity in combination with a threshold
to determine whether the individual replicates past behavior or maximizes preferences
while in Gilboa and Schmeidler (1995) preferences are always maximized. Thus, as
section 7 suggests, case-based decision theory can be ingrained in the more general
structure proposed here. The model in Gilboa and Schmeidler (1995) can be seen as
a particular way of making conscious decisions. Nevertheless, both models agree on
the importance of analogies for human behavior. In Gilboa and Schmeidler (1995)
analogies are used to find the action that maximizes individual utility in a world
without priors, here analogies can be potentially dangerous because they determine
whether the decision maker thinks about his choices or just chooses automatically
without consciously analyzing the problem.
We would like to stress that even if the behavioral model we propose is new,
the idea that observed behavior can be the outcome of the interaction between two
different selves is not novel and it dates back at least to Strotz (1955). Strotz kind of
models, such as Laibson (1997), Gul and Pesendorfer (2001) or Fudenberg and Levine
(2006), are different from the behavioral model we introduce here, since they represent
the two selves as two agents with different and conflicting preferences, usually longrun vs short-run preferences.4 In our approach however, the two selves are inherently
different one from the other. One uses analogies to deal with the environment in which
the decision maker acts, while the other one uses a preference relation to consciously
choose among the alternatives available to the decision maker. Furthermore, which
self drives a particular decision problem depends on the problems that have been
experienced and how similar they are with the present one and thus, it does not
depend on whether the decision is affecting the present or the future. Nevertheless,
we do not exclude the possibility that the fact that a decision affects the present or
the future has some kind of influence on how analogies are made.
Finally, it is important to notice that the preference revelation strategy we use in
4
In some models the difference between the two selves comes from the fact that they have different
information. See for example Cunningham (2015) that proposes a model of decision making where
the two selves hierarchically aggregate information before choosing an alternative. Other papers use
more than two selves to rationalize individual choices, but still all selves are represented by different
preference relations, see for example Kalai et al. (2002), Manzini and Mariotti (2007) and Ambrus
and Rozen (2015).
5
the paper agrees with the one used in Bernheim and Rangel (2009). They analyze the
same problem of eliciting individual preferences from behavioral datasets, and they
do this in two stages. In a first stage they take as given the welfare relevant domain,
that is the set of observations from which individual preferences can be revealed, and
then in a second stage they analyze the welfare relevant observations and propose a
criterion for the revelation of preferences that does not assume any particular choice
procedure to make welfare judgments.5 Even if similar, our approach differs in two
important aspects. First, by modeling conscious and intuitive choices, we propose a
particular method to find the welfare relevant domain, i.e. the algorithm highlighting
a set of conscious choices. Second, by proposing a specific choice procedure, we use
standard revealed preference analysis on the relevant domain, thus our method, by
being behaviorally based, is less conservative for the elicitation of individual preferences. In this sense, our stance is also similar to the one proposed in Rubinstein and
Salant (2012), Masatlioglu et al. (2012) and Manzini and Mariotti (2014) that make
the case for welfare analysis based on the understanding of the behavioral process
that generated the data.
3
Dual Decision Processes
3.1
The Model
Let X and E be two sets. The decision maker (DM) faces at time t a decision problem
(At , et ) with At ⊆ X and et ∈ E. The set of alternatives At that is available at time t,
and from which the DM has to make a choice, is called the menu. An alternative is any
element of choice like consumption bundles, lotteries or streams of consumption. The
environment et is a description of the possible characteristics of the problem that the
DM faces at time t. As highlighted below, an environment can be the menu, the set
of attributes of alternatives in the menu, the framing, etc. We denote by at ∈ At the
chosen alternative at time t. With a little abuse of the notation, we refer to the couple
5
Notice that Apesteguia and Ballester (2015) propose an approach to measure the welfare of
an individual from a given dataset that is also choice-procedure free. They do so by providing a
model-free method to measure how close actual behavior is to the preference that best reflects the
choices in the dataset.
6
formed by the decision problem (At , et ) and the chosen alternative at as observation
t. We denote the collection of observations in the sequence {(At , et , at )}Tt=1 as D, i.e.
D = {1, ..., T }. Notice that the same menu or environment can appear more than
once in the sequence.
A chosen alternative is the outcome of a two stage choice procedure that describes
the DM and that formalizes the duality of automatic and conscious choices. Formally,
let σ : E × E → [0, 1] be the similarity function. The value σ(e, e0 ) measures how
similar environment e is with respect to e0 . The automatic self is endowed with a
similarity threshold α ∈ [0, 1] that delimits which pairs of environments are similar
enough. Whenever σ(e, e0 ) > α the individual considers e to be similar enough to e0 .
At time t and facing the decision problem (At , et ), the automatic self executes a choice
if it can replicate the choice of a previous period s < t such that σ(et , es ) > α. The
choice is the alternative as chosen in one such period. The maximizing self is endowed
with a preference relation over the set of alternatives. For ease of exposition, we
assume that is a strict order, i.e. an asymmetric, transitive and complete binary
relation, defined over X. At time t, if the maximizing self is activated it chooses the
alternative at that maximizes in At .6 Summarizing:

as for some s < t such that σ(et , es ) > α and as ∈ At ,
at =
the maximal element in A with respect to , otherwise.
t
Three remarks are useful here. First, notice that automatic and conscious decisions
are separated by the behavioral parameter α. In some sense α is summarizing the
cost of using the cognitive demanding system. The higher the cost, the lower the
threshold. Thus, parameter α captures individual heterogeneity as preferences do.
Notice that while the similarity function has been widely studied in cognitive sciences,
e.g. Tversky (1977), Medin et al. (1993) and Hahn (2014), the cognitive costs of
activating conscious decisions processes are still an unknown, thus the method we
propose in section 4 can be seen as a first attempt of identifying from observed
6
The idea that conscious behavior comes from the maximization of a preference relation is a
simplification we use to focus the analysis on the main novelties of the framework presented in this
paper. For a more detailed discussion regarding this point, see section 7.
7
behavior the interval in which such costs should lie, given the similarity function.
As a second remark, notice that we are describing a class of models because we do
not impose any particular structure on replicating behavior. We do not specify which
alternative would be chosen when more than one past choice can be replicated. Many
different behaviors can be part of this class, e.g choosing the alternative that was chosen in the most similar past environment or choosing the alternative that maximizes
the preference relation over the ones chosen in similar enough past environments, etc.
All the analysis that follows is valid for the class as a whole.
As a final remark, for illustrative purposes, we propose here some examples of
environments that are relevant for economic applications and a possible similarity
function that can be used in such cases.
Environments as Menus: In many economic applications it seems sensible to see
the whole menu of alternatives, e.g. the budget set, as the main driver of analogies.
That is, E could be the set of all possible menus and two decision problems are perceived as similar as their menus are. In this framework, E = 2X .
Environments as Attributes: Decision makers many times face alternatives that
are bundles of attributes. In those contexts, it is reasonable to assume that the attributes of the available alternatives determine the decision environment. If A is the
set containing all possible attributes, then E = 2A .
Environments as Frames: We can think of the set E as the set of frames or ancillary conditions as described in Salant and Rubinstein (2008) and Bernheim and
Rangel (2009). Frames are observable information that is irrelevant for the rational assessment of alternatives, for example how the products are disposed on a shelf.
Every frame can be seen as a set of irrelevant features of the decision environment.
Thus, if the set containing all possible irrelevant features is F , we have E = 2F .
In all the previous examples it is natural to assume that the similarity function relates different environments depending on their commonalities and differences. For
example, σ(e, e0 ) =
|e∩e0 |
,
|e∪e0 |
that is, two environments are more similar the more char-
acteristics they share relative to all the characteristics they have. Such function is
just a symmetric specification of the more general class considered in Tversky (1977).
8
Although, it is sometimes not possible to have all the information regarding the similarity function, a case we analyze in the appendix, from now on we take E and σ as
given.
3.2
Cognitive Foundations of the Model
This section maps recent findings in cognitive sciences into the model and shows
which simplifying assumptions were made. The model tries to formalize the evidence
concerning the activation of non-deliberative processes. With this objective in mind,
first we present which mental state makes unconscious processes play an important
role in individual decision making and then we study how such state arises.
Cognitive sciences have discovered that unconscious processes are extremely context dependent. The activation of non deliberative cognition depends on the characteristics of the environment. In particular, the studies on fluency, i.e., the subjective
experience of ease associated with a mental process or a particular situation, highlight
that the influence of unconscious processes is stronger in those contexts that trigger
a higher sensation of cognitive ease.7
The studies surveyed by Oppenheimer (2008), one of the leading scholars in fluency
research, show that in fluent, familiar, situations individuals decisions tend not only
to be less conscious, but also less coherent and more prone to errors and biases.8 Here
we propose a model of fluency that is based on the formalization of what is usually
considered its source, i.e., priming.
Paraphrasing Oppenheimer (2008), the interpretation of a fluent experience relies
7
Fluency of a situation is seen as the main determinant of automatic or fast thinking as defined
in Dual Process Theory. See Schneider and Shiffrin (1977), Evans (1977), McAndrews et al. (1987),
Evans (1989), Reber (1989), Epstein (1994) and Evans and Over (1996) for the development of Dual
Process Theory as described in Evans and Frankish (2009) and Kahneman (2011) that uses much
of the evidence presented here and that is related to the model proposed.
8
For example, Alter et al. (2007) show that individuals choosing in a fluent environment tend
to do worst in the Cognitive Reflection Test and in solving logical syllogisms than individuals in a
disfluent situation. They also show that individuals in a fluent setting tend to evaluate consumer
products by placing less weight on their objective characteristics. In another study, Hernandez and
Preston (2013) show that putting people in a disfluent environment disrupts the confirmation bias.
Another example comes from the study conducted by Costa et al. (2014). They show that students
put in a fluent environment in which assignments are written in their mother tongue tend to make
more biased decisions than children put in a disfluent one where assignments are written in a foreign
language.
9
on the unconsciously perceived relationship between past experiences and the current
context. Such relationship is analyzed by priming that studies the influence of unconscious or implicit memory on behavior through environmental cues. As defined
in Tulving and Schacter (1990), priming is an unconscious change in the ability to
identify or produce an item as a result of a specific prior encounter with the item.
Priming creates a sense of familiarity and ease that makes the environment fluent
and behavior automatic and coherent with the context. The behavioral model proposed here, even in its simplicity, captures some of the main characteristics of priming
in a stylized way.
Bargh (2005), one of the major contributors to the study of priming, highlights
the following characteristics of the phenomenon:
1. It is unconscious.
2. It is based on the conceptual representation of the environment.
3. It determines behavior.
4. It is costly to control.
5. It has a long term effect on behavior and it is not affected by age.
The remainder of the section explains how the model incorporates these points.
Regarding point 1, the evidence presented in Bargh (2005) shows that priming is
completely unconscious. Quoting Bargh:
“. . . individual’s behavior is being controlled by external stimuli not by his or her
consciously accessible intentions or acts of will. . . action tendencies can be activated
and triggered independently and in the absence of individual’s conscious choice or
awareness of those causal triggers.”
Neurological evidence strengthens even more this finding. Many studies have shown
a decrease in neural activity when an individual faces an environment that primes
behavior, a phenomenon known as repetition suppression. The decrease of activity
affects in particular neurons that are usually associated with conscious recognition of
10
the stimuli, while neurons associated with unconscious recognition are not affected as
much. This physiological reaction points to the fact that recognition of environmental cues precedes any conscious control of behavior. The model captures this idea by
assuming that the automatic self acts in the first stage of the choice procedure.
The evidence presented by Bargh (2005) shows that priming is effective even if the
environment slightly changes with respect to the reference one, that is, even if some
features have changed.9 As stated in point 2, priming is based on a conceptual representation of experienced environments. In the model, an environment can influence
present behavior because it is similar enough to, i.e., it shares the same conceptual
representation of, the environment that primes behavior.
Point 3 is the most important from a behavioral perspective and it is a defining
characteristic of priming. Again, quoting Bargh:
“. . . an individual’s behavior can be directly caused by the current environment. . . this
behavior can and will unfold without the person being aware of its external determinant.”
This idea is the basis of priming. The most radical examples in this direction come
from subjects with prefrontal cortex dysfunctions, a critical region for conscious and
controlled behavior. Many studies, e.g., Lhermitte (1986), Ghika et al. (1995) and Bisio et al. (2012), have shown that these individuals can be made to drastically change
their behavior just by manipulating the environment. The reaction is coherent with
the context priming behavior. Such extreme dependency on the environment, called
environmental dependency syndrome, is a radical manifestation of priming, as Bargh
(2005) states, given it activates the same areas of the brain.10 This complex idea is
simplified in the paper by means of replication of past behavior from familiar environments.
9
See also Higgins and Bargh (1987) and Bargh et al. (2001) for similar discussions.
It is interesting to note that healthy individuals unconsciously experiencing priming try to
consciously justify their behavior ex-post. That is, the influence of priming on behavior is much
more subtle than what it seems at a first glance. Individuals can consciously justify their behavior
ex-post without acknowledging the influence of the environment in the moment preceding the action.
See Bargh and Morsella (2008) for a defense of the idea that action precedes reflection.
10
11
Finally, points 4 and 5 are important to understand why we assume that (i) α is
an individual characteristic and (ii) memory is perfect. Regarding the first point, the
similarity threshold tries to catch the idea that overcoming fluency to take control of
one’s own actions is costly and depends on individual cognitive characteristics. With
respect to the second point, the assumption of perfect memory is a simplification
of the fact that, as highlighted by the evidence in Bargh (2005) and also in Duhigg
(2012), behavior can be primed by experiences stored in implicit memory that can
be forgotten by explicit, or conscious, memory. For example, there is evidence that
patients with amnesia are affected by priming as strongly as any other individual.
Moreover, it seems that priming effects do not deteriorate with age, hence pointing
to the fact that implicit memory is not affected by age as explicit memory is. Notice
however, that this assumption can be weakened without affecting the results.
To conclude, the model captures important aspects of fluency and priming in a
simple way while keeping the main insights of environmental influence on behavior.
3.3
An example
This section provides an example to illustrate the behavioral process we are modeling.
Even if the example is abstract, it shows in a very direct way how the model works.
For more concrete applications of the model to interesting economic settings, please
refer to section 6.
Let X = {1, 2, 3, ..., 10}. We assume that environments are menus, i.e. E is the set
of all subsets of X and we assume that σ(A, A0 ) =
|A∩A0 |
.
|A∪A0 |
Suppose that the automatic
self is described by α = .55 and that the preference 1 2 3 · · · 10 describes
the maximizing one. We now explain how our DM makes choices from the following
list of ordered menus:
t=1
t=2
3, 4, 5, 6 1, 2, 3, 4, 5, 6
t=3
t=4
t=5
1, 3, 4, 7 2, 4, 7
1, 3, 6
t=6
t=7
1, 2, 3, 4 2, 4, 8
t=8
2, 4, 8, 9, 10
In the first period, given the DM has no prior experiences, the maximizing self
is going to be active. Thus, the choice comes from the maximization of preferences,
12
that is, a1 = 3. Then, in the following period, given that σ(A2 , A1 ) =
4
6
> .55, the
automatic self is active and so we would observe a replication of past choices, that is,
a2 = 3. Period 1’s environment makes period 2’s one fluent, hence priming behavior.
Now, in period 3, notice that the similarity between A3 and A2 or A1 is always below
the similarity threshold and this makes the maximizing self active. The preference
relation is maximized and so a3 = 1. A similar reasoning can be applied for the
fourth and fifth periods to see that a4 = 2 and a5 = 1. Then, in period six, the
automatic self is active given that σ(A6 , A3 ) =
3
5
> .55, leading to a6 = 1. In period
seven, given no past environment is similar enough, the maximizing self is active and
so a7 = 2. Finally, in the last period the automatic self is active again given that
σ(A8 , A7 ) =
3
5
> .55 and so behavior will be replicated, i.e. a8 = 2.
One may wonder what an external observer would understand from this choice
behavior. Section 4 addresses this point
4
The Revealed Preference Analysis of Dual Decisions
In this section, we discuss how to recognize which observations were automatically
or consciously generated in a DD process. This information is crucial to elicit the
unobservables in the model that are the sources of individual heterogeneity, that
is the preference relation and the similarity threshold. As previously discussed in
Section 3, the similarity function is taken as given, while the similarity threshold
α is elicited from observed behavior. Notice that, the similarity function and the
similarity threshold define a binary similarity function that is individual specific.
Thus, by separating the similarity function from the similarity threshold we are able
to associate individual heterogeneity to a parameter related with individual cognitive
costs without having too many degrees of freedom to properly run the technical
analysis.
It is easy to recognize a set of observations that is undoubtedly generated by the
maximizing self. Notice that all those observations in which the chosen alternative
was never chosen in the past must belong to this category. This is so because, as no
13
past behavior has been replicated, the automatic self could not be active. We call
these observations new observations.
In order to identify a first set of observations automatically generated, notice that
the maximizing self being rational, it cannot generate cycles of revealed preference.
As it is standard, a set of observations t1 , t2 , . . . , tk forms a cycle if ati+1 ∈ Ati ,
i = 1, . . . , k − 1 and at1 ∈ Atk , where all chosen alternatives are different. Given
the previous reasoning, for every cycle there must be at least one observation that
is automatically generated. Intuitively, the one corresponding to the most familiar
environment should be a decision mimicking past behavior.
The unconditional familiarity of observation t is
f (t) =
max σ(et , es ).
s<t,as ∈At
Whenever there is no s < t such that as ∈ At , we say f (t) = 0.
That is, unconditional familiarity measures how similar observation t is to past
observations from which behavior could be replicated, i.e., those past decision problems for which the chosen alternative is present at t. Then, we say that observation
t is a most familiar in a cycle if it is part of a cycle of observations, and within it, it
maximizes the value of the unconditional familiarity.
The major challenge is to relate pairs of observations in a way that allows to
transfer the knowledge of which self generated one of them to the other. In order to
do so, we introduce a second measure of familiarity of an observation t, that we call
conditional familiarity. Formally,
f (t|at ) = max σ(et , es ).
s<t,as =at
Whenever there is no s < t such that as = at , we say f (t|at ) = 0.
That is, conditional familiarity measures how similar observation t is with past
observations from which behavior could have been replicated, i.e., those past decision
problems for which the choice is the same as the one at t. The main difference between
f (t) and f (t|at ) is that the first one is an ex-ante concept, i.e., before considering
the choice, while the second one is an ex-post concept, i.e., after considering the
14
choice. Our key definition uses these two measures of familiarity to relate pairs of
observations.
Definition 1 (Linked Observations) We say that observation t is linked to observation s, and we write t ∈ L(s), whenever f (t|at ) ≤ f (s). We say that observation t
is indirectly linked to observation s if there exists a sequence of observations t1 , . . . , tk
such that t = t1 , tk = s and ti ∈ L(ti+1 ) for every i = 1, 2, . . . , k − 1.
The definition formalizes the key idea behind the algorithm. Observation t is linked to
observation s if its conditional familiarity is below the unconditional familiarity of s.
As explained below, once an observation is categorized as consciously or automatically
generated, it is through its link to other observations that such knowledge can be
extended.
Denote by DN the set of all observations that are indirectly linked to new observations and by DC the set of all observations to which most familiar observations in
a cycle are indirectly linked. The binary relation determined by the concept of linked
observations is clearly reflexive, thus, new observations and most familiar observations in a cycle are contained in DN and DC respectively. We are ready to present
the main result of this section. It establishes that observations in DN are generated
by the maximizing self, while observations in DC are generated by the automatic
self. As a consequence, it guarantees that the revealed preference of observations in
DN is informative with respect to the preferences of the individual and, moreover, it
identifies an interval in which the similarity threshold has to lie. Before stating the
proposition, it is useful to highlight that x is revealed preferred to y for a set of observations O, i.e., xR(O)y, if there is a sequence of different alternatives x1 , x2 , . . . , xk
such that x1 = x, xk = y and for every i = 1, 2, . . . , k − 1 ∈ O, it is xi = at and
xi+1 ∈ At for some t.
Proposition 1 For every collection of observations D generated by a DD process:
1. all observations in DN are generated by the maximizing self while all observations in DC are generated by the automatic self,
2. if x is revealed preferred to y for the set of observations DN , then x y,
15
3. max f (t) ≤ α < min f (t|at ).
t∈DN
t∈DC
Proof. We start by proving the statement regarding conscious observations. Trivially,
new observations must be generated by the maximizing self since they cannot replicate
any past behavior. Consider an observation t ∈ DN . By definition, there exists a
sequence of observations t1 , t2 , . . . , tn with t1 = t, f (ti |ati ) ≤ f (ti+1 ) for all i =
1, 2, ..., n − 1 and tn being new. We prove that t is generated by the maximizing
self recursively. We know that tn is consciously generated. Now assume that tk is
generated by the maximizing self and suppose by contradiction that tk−1 is generated
by the automatic one. From the assumption on tk , we know that f (tk ) ≤ α. From the
assumption on tk−1 , we know that f (tk−1 |atk−1 ) > α, which implies f (tk−1 |atk−1 ) >
f (tk ), a contradiction with the hypothesis. Hence, tk−1 is also generated by the
maximizing self, and the recursive analysis proves that observation t is also consciously
generated.
We now prove the statement regarding automatic observations. Consider first an
observation t which is a most familiar in a cycle and assume by contradiction that it
is generated by the maximizing self. Then, f (t) ≤ α. By definition of most familiar in
a cycle, it must be f (s) ≤ α for every s in the cycle, making all decisions in the cycle
being generated by the maximizing self. This is a contradiction with the maximization
of a preference relation. Consider now an observation t ∈ DC . By definition, there
exists a sequence of observations t1 , t2 , . . . , tn with tn = t, f (ti |ati ) ≤ f (ti+1 ) for all
i = 1, 2, ..., n − 1 and t1 being a most familiar in a cycle. We proceed recursively
again. Since t1 is generated by the automatic self, we have f (t1 |at1 ) > α. Now
assume that tk is generated by the automatic self and suppose by contradiction that
tk+1 is generated by the maximizing one. We then know that f (tk |atk ) > α ≥ f (tk+1 ),
which is a contradiction concluding the recursive argument.
For the revelation of preferences part, since DN can only contain observations
generated by the maximizing self, it is straightforward to see that the revealed information from such a set must respond to the preferences of the DM. Regarding α,
notice that since observations in DN are generated by the maximizing self, we know
that maxt∈DN f (t) ≤ α and also that, since observations in DC are generated by the
automatic self, α < mint∈DC f (t|at ), which concludes the proof.
16
To understand the reasoning behind Proposition 1, consider first an observation
t that we know is new, and hence consciously generated. This implies that its corresponding environment is not similar enough to any other previous environment.
In other words, f (t) ≤ α. Then, any observation s for which the conditional familiarity is less than f (t) must be generated by the maximizing self too. In fact,
f (s|as ) ≤ f (t) ≤ α implies that no past behavior that could have been replicated in s
comes from an environment that is similar enough to the one in s. Thus, any observation linked with a new observation must be generated by the maximizing self. It is
easy to see that this reasoning can be iterated, in fact, any observation linked with an
observation generated by the maximizing self must be generated by the maximizing
self too.
Similarly, consider a most familiar observation in a cycle t, that we know is automatically generated. Any observation s for which the unconditional familiarity is
greater than the conditional familiarity of t must be generated by the automatic self
too. In fact, we know that α < f (t|at ) because t is generated by the automatic self.
Then, any observation s to which t is linked has an unconditional familiarity above
α, which implies that some past behavior could be replicated by the automatic self,
and so such observation must be generated by the automatic self too. Again, the
reasoning can be iterated. Thus, we can start from a small subset of observations
undoubtedly generated either automatically or consciously, inferring from there which
other observations are of the same type. We use the example of section 3.3 to show
how the algorithm works.
Algorithm: Example of Section 3.3
Suppose that we observe the decisions made by the DM in the example of section 3.3,
without any knowledge on his preferences or similarity threshold α. The following
table summarizes the different observations.
17
t=1
t=2
t=3
t=4
t=5
t=6
t=7
t=8
3, 4, 5, 6
1, 2, 3, 4, 5, 6
1, 3, 4, 7
2, 4, 7
1, 3, 6
1, 2, 3, 4
2, 4, 8
2, 4, 8, 9, 10
a1 = 3
a2 = 3
a3 = 1
a4 = 2 a5 = 1
a6 = 1
a7 = 2
a8 = 2
We can easily see that the only new observations are observations 1, 3 and 4, and
hence we can directly infer that the corresponding choices were conscious.
We can go one step further and consider observation 5. From observed behavior
we cannot understand whether the choice comes from maximization of preferences or
the replication of past behavior in period 3. Nevertheless, the choice was conscious in
period 3 and one can easily see that f (5|a5 ) =
2
5
≤
3
7
= f (3), making observation 5
linked with observation 3 and according to Proposition 1, revealing it was conscious
too.
Consider now observation 7. We cannot directly link observation 7 to either observations 1, 3 or 4, because f (7|a7 ) =
1
2
> max{f (1), f (3), f (4)}. However, we
can indirectly link observation 7 to observation 3 through observation 5, because
f (7|a7 ) =
1
2
≤
1
2
= f (5), thus making 7 an element of DN . No other observation
is indirectly linked to observations 1, 3 or 4 and hence, DN = {1, 3, 4, 5, 7}. The
method rightfully excludes all automatic choices from DN .
The example presents inconsistencies in the revealed preference. Observation 3 and
6 are both in conflict with observation 2. That is, observations 2 and 3 and 2 and 6
form cycles. Then, noticing that max{f (2), f (3)} = f (2) and that max{f (2), f (6)} =
f (2) = f (6) we can say that observations 2 and 6 are generated by the automatic self
thanks to Proposition 1, given they are most familiar in a cycle.
But then, notice that observation 6 is linked to observation 8 given that f (6|a6 ) =
3
5
≤ f (8) =
3
5
revealing that the latter must have been automatically generated too.
Thus, we get DC = {2, 6, 8} that were the observations rightfully excluded from DN .
No decision made by the maximizing self has been cataloged as automatic.
The modified revealed preference exercise helps us determine that alternative 1 is
better than any alternative from 2 to 7, alternative 3 is better than any alternative
from 4 to 6, and alternative 2 is better than alternatives 4, 7 and 8 as it is indeed
the case. The value of the similarity threshold α by Proposition 1 can be correctly
determined to be in the interval [0.5, 0.6) thanks to the information retrieved from
18
observations 7 and 8 respectively.
DN contains new observations and those indirectly linked to them.11 It may be
the case that some observations consciously generated are not linked to any observation in DN , hence making DN a proper subset of the set of all consciously generated
observations. For this reason, nothing guarantees that D \ DN are observations automatically generated and hence, Proposition 1 needs to show how to dually construct
a set of automatic decisions DC . Nonetheless, if the observations are rich enough, it is
possible to guarantee that DN and DC contain all conscious and automatic decisions,
respectively.12 More importantly, notice that Proposition 1 relies on one important
assumption, that is, the collection of observations is generated by a DD process. The
following section addresses this issue.
5
A Characterization of Dual Decision Processes
In Section 4 we showed how to elicit the preferences and the similarity threshold of
an individual that follows a DD process. Here, building upon the results of that
section, we provide a necessary and sufficient condition for a set of observations to be
characterized as a DD process with a known similarity function. In other words, we
provide a condition that can be used to falsify our model.
From the construction of the set DN , we understand that a necessary condition
for a dataset to be generated by a DD process is that the indirect revealed preference
we obtain from observations in DN , i.e. R(DN ), must be asymmetric. It turns out
that this condition is not only necessary but also sufficient to represent a sequence of
decision problems as if generated by a DD process. One simple postulate of choice
characterizes the whole class of DD processes.
Axiom 1 (Link-Consistency) A sequence of observations {(At , et , at )}Tt=1 satisfies
Link-Consistency if, in every cycle there is at least one observation not indirectly
linked to a new observation.
11
12
DN is never empty because it always contains the first observation.
Material available upon request.
19
This is a weakening of the Strong Axiom of Revealed Preference. In fact it allows
for cycles but only of a particular kind. Preferential information gathered from observations in DN cannot be inconsistent. The next theorem shows that such condition
is indeed necessary and sufficient to characterize DD processes with known similarity.
Theorem 1 A sequence of observations {(At , et , at )}Tt=1 satisfies Link-Consistency
if and only if there exist a preference relation and a similarity threshold α that
characterize a DD process.
Proof. Necessity: If D is generated by a DD process, then it satisfies Link-Consistency
as explained in the text.
Sufficiency: Now suppose that D satisfies Link-Consistency. We need to show that
it can be explained as if generated by a DD process. Notice that Link-Consistency
implies that the revealed preference relation defined over DN , i.e. R(DN ), is asymmetric. In fact, asymmetry of R(DN ) means that it is not possible to construct cycles
composed by observations in DN . By standard mathematical results, we can find a
transitive completion of R(DN ), call it . By construction, all decisions in DN can
be seen as the result of maximizing over the corresponding menu.
Define α = maxt∈DN f (t). Notice that by definition of DN , there is no observation
s∈
/ DN such that f (s|as ) ≤ f (t) for some t ∈ DN . This implies that for all s ∈
/ DN ,
f (s|as ) > α, so, for all of them, it is possible to find a preceding observation they
would seem to replicate. In particular, the one defining f (s|as ).
Thus, we can represent the choices as if generated by an individual with preference
relation and similarity threshold α.
The theorem is saying that Link-Consistency makes possible to determine whether
the DM is following a DD process or not. In particular, when the property is satisfied,
we can characterize the preferences of the DM with a completion of R(DN ) which is
asymmetric thanks to Link-Consistency and use the lower bound of α as described
in Proposition 1 to characterize the similarity threshold. In fact, by construction, for
any observation t outside DN it is possible to find a preceding observation that can be
replicated, i.e., the one defining f (t|at ).13 In the Appendix we show how the algorithm
13
Notice that Link-Consistency implies that there have to be no cycles among new observation,
20
would work if the knowledge of similarity comparisons is partial. Finally, notice that
we do not assume any particular structure for the sequence of observations we use as
data and hence, the characterization of preferences does not have to be unique, even
when the similarity is known.14
6
Dual Decisions Processes: General Implications
This section presents some applications of the model for various settings. The aim
is to show that the model provides a simple possible explanation for some puzzling
empirical regularities that have been highlighted in different fields. It does so very
directly, without the need of being tailored too much to a particular application. In
particular, only the interpretation of E and X will change. In fact, for all settings,
we assume that E and X are subsets of some real space. Furthermore, for any e ∈ E,
we assume there exist an optimal action x(e) ∈ X, that is different for different
environments, that is, for any e, e0 ∈ E we have x(e) 6= x(e0 ). Moreover, let the
similarity function be σ(e, e0 ) =
1
1+d(e,e0 )
where d is the euclidean distance between
vectors e and e0 .
The key idea on which the remainder of the section builds is that the model formalizes a novel way of understanding the coexistence of sticky and adaptive behavior.
In section 6.1 the model is used to reconcile some puzzling evidence on doctors’ behavior, in section 6.2 the model helps understanding new evidence on sticky pricing and
finally in section 6.3 the model provides a possible interpretation of some empirical
regularities highlighted in financial markets.
6.1
Dual Decision Processes and Treatments Prescription
Suppose doctors are described by a DD process. Let E ⊂ Rn be the compact set
of symptom profiles. That is, e ∈ E describes a unique list of symptoms that a
patient could display. Furthermore, let X ⊂ Rn be the set of possible treatments.
a concept independent of the particular similarity function chosen, thus making the model always
falsifiable.
14
Material providing a full characterization of the procedure with rich enough data is available
upon request.
21
The automatic self makes analogies between the symptom profile of a patient visiting
the doctor and the symptoms profiles the doctor has seen in the past. If there is one
that is similar enough, the doctor automatically assigns to the new patient the same
treatment he gave in the past. Otherwise, the maximizing self is activated and the
doctor prescribes the optimal treatment.
Our doctor has a memory composed by past cases, (e, x) ∈ M ⊆ E × X, with the
interpretation that the doctor has seen a patient with symptoms e and has given him
treatment x. Let E M be the projection of M in E. This means that E M contains all
those symptom profiles the doctor has experienced.
This model clearly oversimplifies the complexities of treatment prescription. First,
it does not consider the uncertainty inherent to clinical decision making. Second, it
does not include the cost of diagnosis nor the possibility of mistakes when using
the maximizing self, hence abstracting from the positive effect of experience through
learning. These assumptions would worsen the exposition without adding to the
general results.
The first implication of the model is immediate.
Remark 1 For every new patient, doctors described by a DD process, either categorize the patient with past similar ones or they give him a tailored treatment.
This is in line with the evidence shown in Frank and Zeckhauser (2007). Their
results highlight the fact that doctors cluster some patients giving them the same
treatment while individually studying other cases. Moreover, when prescribing categorized treatments doctors are faster, i.e., the visits are shorter. This might be an
indication that the cognitive process underlying treatment prescription when patients
are categorized is automatic, that is, it is less cognitively costly for the doctor. Moreover, Frank and Zeckhauser (2007) show that such behavior is far from optimal and
call for a behavioral explanation of the phenomenon.
A second, direct, implication is that the automatic self causes repetition of prescription behavior, or more simply habits, to emerge.
Remark 2 Doctors described by a DD process show habits in treatment prescription.
22
Any time the automatic self is activated past treatments are prescribed again to
new patients. This implies that for diseases our doctor has already treated, even
if new treatments are discovered, new patients affected by the same diseases will
receive the outdated and suboptimal treatment. This is in line with the evidence.
For instance Hamann et al. (2004) show that some psychiatrists continue to prescribe
first generation anti-psychotics even though last generation ones are available.
Obviously, this phenomenon could be due to learning or uncertainty aversion on
the side of doctors, that is, doctors might prescribe more those drugs they have
previously prescribed and for which they know the effects. This kind of explanation,
plausible in many instances, is in contrast with the evidence shown in Hellerstein
(1998). She examines prescription behavior of physicians when deciding between
generics and branded drugs. Even if the U.S. Food and Drug Administration refers
to generic drugs as “chemical carbon copies” of branded drugs, doctors that have
prescribed branded drugs tend to continue to do so with new patients. This behavior,
at odds with learning, might be due to trust. Doctors and/or patients tend not to
trust generics. Nevertheless, the latter interpretation cannot be easily reconciled with
the evidence provided in Hamann et al. (2004) and Shrank et al. (2011) that show
that this kind of inconsistent behavior seems to be correlated with the age of the
doctor. Older doctors tend to prescribe older, e.g., branded, drugs. The next remark
shows that this empirical regularity can be reproduced in our model.
Remark 3 Suppose that patients arrive to the doctor following a continuous time
independent distribution F with full support on E. As the memory of the doctor
increases, the probability of using the automatic self increases.
Proof. First, call N 1−α (e) the
α
1−α
-neighborhood
α
of e ∈ E, that is
1−α
0
0
N 1−α (e) = {e ∈ E|σ(e, e ) > α} = e ∈ E|d(e, e ) <
α
α
0
0
Second, let N (E M ) be the union of
1−α
-neighborhoods,
α
23
i.e., N 1−α of all the cases
α
in E M intersected with E. Formally,
N (E M ) =
[
N 1−α (e) ∩ E,
α
e∈E M
that is N (E M ) is the set containing all the cases in the doctor’s memory plus all
those that would be considered similar enough by the doctor. Notice that this set is
increasing in E M .
Finally, notice that the probability that a new patient is considered similar enough
to one of the cases in the memory, i.e., the probability of activating the automatic
self, is F (N (E M )) which is increasing in its argument and the result follows.
At first glance Remark 3 seems to suggest that older doctors should incur stronger
habits, in line with the evidence. Nevertheless, the proposition is really saying that
this behavior is not due to age per se but to the experience of doctors of which age is
a proxy. Thus, in theory, two doctors of different ages but with the same experience
over one category of diseases should have the same prescription behavior. This is
something to test in the data that we leave for further research.
The experience of the doctor is clearly central in the model. Doctors’ behavior is
path dependent, in the sense that, whenever the automatic self is activated, which
treatment the doctor prescribes heavily depends on what he has seen in the past.
This creates heterogeneity in behavior in line with the evidence that Chandra et al.
(2011) present about variation in treatment prescription among different regions in the
U.S.. They show that such variation cannot be completely explained by demand and
supply factors, and so they call for a behavioral theory to account for the remaining
variation. The model proposed here might be a possible explanation to consider. This
problem is central in health economics because it translates in an impossibility for
the government to control the expenditures in medical services across regions.
Finally, notice that the kind of implications analyzed in this section are not limited
to doctors’ behavior. Experts in general, like managers, military officers or political
analysts, can be described in this way. The trade-off between positive and negative
effects of expertise have to be explored more deeply. As Kahneman (2002) highlights,
experts show pattern of behavior that are inconsistent and heavily affected by biases
24
like confirmation bias and availability heuristics that are intertwined with the way
analogical thinking is formalized here.
6.2
Dual Decision Processes and Prices
Suppose retailers are described by a DD process. Let E ⊂ R be a compact set of
perfect signals about the demand of the product sold. That is, e ∈ E is a signal that
perfectly describes the state of the market for the retailer. Moreover, let X ⊆ R be
the set of possible prices.
Given this structure it is immediate to see that pricing can be sticky. Suppose
retailers receive in period t the signal et about the state of the market and that signals are drawn from a continuous and time invariant distribution F with full support
on E. The automatic self compares et with any market environment previously encountered. If the market is similar enough with what has been experienced, then the
automatic self is active and pricing is replicated. On the other hand, if environments
are dissimilar enough, prices are optimally adapted to the market. This is highlighted
in the following remark.
Remark 4 If retailers are described by a DD process, prices are not always optimally
adapted to the market environment.
There is ample evidence that prices are sticky. The usual explanation first proposed
by Sheshinski and Weiss (1977) is that adapting prices to the market has a cost, called
menu cost, making it suboptimal to constantly optimally modify prices. Prices are
changed only when the benefits of the update compensate the costs. In a dynamic
environment, the longer the inaction the higher the benefits of updating the price
because the old pricing rule gets farther away from the optimum. Thus in general,
the probability that a price adapts to the market situation should be higher the longer
the period of inaction.
Here we propose a different channel that gives opposite predictions.
Remark 5 Suppose a retailer described by a DD process has not changed the price
for T periods, then, ceteris paribus, the probability of changing the price in period
T + 1 is decreasing in T .
25
Proof. The mechanism is quite straightforward and it is similar to the one behind
Remark 3. In fact, we use the same notation adopted there. Let MT be the set
containing all signals the retailer has received for the T periods in which the price did
not change. Notice that N (MT ) is increasing with respect to the cardinality of MT
given that E is compact. Moreover, MT is increasing in T . Given that the retailer
did not change the price for T periods, we have only two possible scenarios due to
the assumption that to each signal corresponds a unique and different optimal price.
• MT is a singleton. That is, the retailer received always the same signal.
• MT is not a singleton. Given the retailer sticked to one price this implies that
he received a sequence of signals such that every new signal was similar enough
with some other signal already in MT .
Notice that the first case has probability zero, so it must be that the second one
applies which implies that MT is increasing in T . Thus, given the probability of
changing the price in T + 1 is equal to 1 − F (N (MT )), the result follows.
The intuition behind the result is quite straightforward. If a retailer described by
a DD process does not adapt his price to the market, it can only be because he is
replicating past choices. Thus, the longer he sticks to a price, the more signals he has
experienced that are similar enough to the ones he previously saw. The more time
passes, the less is the probability of receiving a signal not similar enough to one of
the signals already experienced.
Interestingly, this prediction of the model is in line with recent empirical evidence.
Using weekly scanner data from two small US cities, Campbell and Eden (2014) find
that the probability of a nominal adjustment of a price decreases with the age of the
price change. That is, the older is the price change the less probable it is that it will
change again. This result, confirmed with Mexican CPI data by Baley et al. (2016),
is at odds with standard economic modeling of price rigidities as underlined before,
but it can be accommodated with the model proposed here.15
Another empirical regularity that Campbell and Eden (2014) find is that the probability of a nominal adjustment of a price is highest when a store’s price substantially
15
See Baley and Blanco (2016) for an alternative explanation of this phenomenon in which menu
costs are combined with learning.
26
differs from the average of other stores prices and that this happens with relatively
young prices. This fact can be easily explained in our framework when certain conditions hold. If all retailers receive independent signals from the same distribution, an
extreme signal, in the sense of it being unlikely, has a higher probability of being dealt
with consciously thus leading to a price adjustment. Furthermore, it is more probable
that it will be followed by another price adjustment because the successive signal is
more likely to be less extreme hence activating the conscious self or the automatic
self replicating a more familiar pricing decision.
A subtle implication of the model is that the more volatile is the market environment, the more frequently prices should be adjusted because it is higher the
probability of facing environments different one from the other. This is in line with
the evidence shown in Bils and Klenow (2004). The authors find that stickiness of
prices varies greatly between markets and that in particular oil and commodities
markets, that are usually quite volatile, tend to be the ones showing lower levels of
stickiness.
Finally, the mechanism described here might also be used to study bank managers reacting in a similar way to changes in monetary policy or, as the next section
highlights, traders reacting to new information in financial markets.
6.3
Dual Decision Processes and Financial Markets
Suppose traders are described by a DD process. Let E ⊆ R be the set containing
perfect and public signals regarding the state of the market. For example, in a simple
economy with one risky asset and one riskless asset, E might be the set containing
the perfect signals that inform on the risky asset, e.g., the expected dividends.16
Moreover, let X contain portfolio compositions. In the example, x ∈ X would say
how much to invest on the risky and on the riskless asset.
Suppose traders in every period t receive a public signal et drawn from a publicly
known distribution G. It is evident that not every trader will consider the signal when
choosing the optimal portfolio. Some traders will use the signal while others will stick
to past decisions. Clearly, depending on the distribution of α in the population and on
16
See Cerigioni (2016) for a deeper analysis of this framework.
27
how demand is determined, signals can have different impact on the price of assets.
To be more precise, let α be distributed in the population following a continuous
distribution F with full support on [0, 1], then we can state the following:
Remark 6 Given a history of signals M , for every signal e there exists an αeM such
that:
• 1 − F (αeM ) traders will fully incorporate the signal into their trading decisions.
• F (αeM ) traders will not incorporate the new information and stick to past trading
decisions.
Proof. The reasoning behind the remark is quite straightforward. Trader i considers
the signal e only if
max
σ(e, e0 ) ≤ αi .
0
e ∈M
Thus, let αeM = maxe0 ∈M σ(e, e0 ) and the result follows. In fact, given signals are
public αeM is uniquely determined for the whole population.
Simply put, traders that have a similarity threshold high enough to consciously perceive the new signal will adapt their choices, the others will replicate past choices.
This is in line with the evidence presented in the literature related to underreaction
of prices in financial markets. Since Cutler et al. (1991) and Bernard (1993) much
empirical evidence has been accumulated that shows that markets tend to underreact
to news. That is, information is not fully and immediately incorporated into prices
as the model proposed here predicts.
There is another feature of the model that is worth underlining.
Remark 7 If the market is composed by traders described by a DD process, the random process determining the realization of present and future signals creates two
sources of uncertainty for the demand of risky assets:
• Uncertainty regarding future movements of the fundamentals.
• Uncertainty regarding the fractions of traders that will consciously use new information.
28
While the first source is standard in the literature, the second one is due to the model
analyzed here and it has the potential of increasing the level of uncertainty in a given
market thus increasing its volatility. This is in line with many empirical studies, e.g.
Shiller (1992), and it can give new insights into the equity premium puzzle presented
in Mehra and Prescott (1985). In fact, the authors show that traders, given the
historical return on equities, tend to underinvest in risky assets with respect to what
standard models would predict. To accommodate for the observed trading decisions,
usual models should assume unrealistic levels of risk aversion. Here we propose a
different framework that, by increasing the risk of the economy, might help explain
this kind of evidence.
Notice that the behavior of traders described in this section can also be a good
description of households and consumption decisions. Households described by a DD
process do not always adapt consumption to information concerning permanent income thus creating, on the population level, the aggregate smoothness of consumption
evident in the data, e.g. Carroll et al. (2011), but not predicted by classical models.
Moreover, the model would produce smoothness of aggregate consumption without
abandoning heterogeneity of household behavior. In fact, following Remark 6, some
households optimally adapt to shocks as highlighted by Carroll and Slacalek (2006)
and in contrast with models of habit formation.
7
Final Remarks
Cognitive sciences have highlighted the fact that choices can be divided into two
categories; conscious choices and automatic ones. This hinders greatly the use of
standard economic models that generally do not take into account the consequences
of there being a difference between conscious and automatic choices.
In this paper, we proposed a possible formalization of this duality that allows us
to restore standard revealed preference analysis. In order to do this, we have assumed
that conscious behavior is the maximization of a given preference relation. In some
cases this assumption can be too strong. Inconsistent choices might arise also when
choosing consciously. Fortunately, the framework and analysis we developed here do
29
not depend on the particular conscious behavior that is assumed. In fact, for any
given decision environment analogies determine a partition with two components,
one containing those problems that are similar enough to the reference environment
and another one containing those that are not. Such a partition is based only on one
and simple assumption, automatic choices must come from the replication of past
behavior.
Alternative conscious behaviors are possible. The only element of the formal
analysis that has to be changed is the consistency requirement that needs to be tested
on those problems in DN . Obviously, dually, to find automatic choices we should
analyze violations of such requirements. Similarly, the assumption that automatic
behavior comes from the replication of past behavior is much less restrictive than
what would appear at a first glance. Automatic choices have to be familiar choices.
For example, we can consider cases where the DM stores in his memory not only his
experiences, but also his parents’, siblings’ or friends’ ones thus making the concept
more general than what the literal interpretation of the model would suggest.
Furthermore, the model can be of use for welfare considerations. Obviously, restoring a revealed preferences approach is crucial for welfare analysis. Nevertheless, the
model proposed here highlights something that is usually not considered, i.e., cognitive costs. The upper and lower bound of the interval in which the similarity threshold
lies are defining a lower and upper bound of the cognitive costs of activation of the
maximizing self, respectively. This can help welfare analysis but opens the question
of how to relate such costs to utility that we leave to future research.
Finally, another point deserves further attention. Here we made some simplifying
assumptions in order to develop a new framework encompassing automatic choices. In
particular, we assumed that the similarity function is known and the environments are
given. The former assumption can be weakened as the Appendix shows, nonetheless
it is essential to know something about similarity comparisons. Automatic behavior can be consistent with the maximization of a given preference relation making
the observation of inconsistent behavior not enough to correctly categorize decisions.
Analogies add a layer of complexity to the problem that demands richer data. A
first step in this direction is to address the second assumption previously mentioned,
30
i.e., that environments are known. Understanding what are the important elements
in a decision problem for similarity comparisons, is something crucial for the study
of individual decision making. The model proposed here provides a setting to think
about this issue in a structured way but leaves the question open for future research.
A
Appendix
A.1
Partial Information on the Similarity
In this section we show that our algorithmic analysis is robust to weaker assumptions
concerning the knowledge of the similarity function. In particular, we study the case
in which only a partial preorder over pairs of environments is known, denoted by S.
For example, the Symmetric Difference between sets satisfies this assumption. Such
extension can be relevant in many contexts where it is not possible to estimate the
similarity function. In such cases, it is sensible to assume that at least some binary
comparisons between pairs of environments are known. Coming back to the example
we use in the introduction of the paper, we might not know how the DM compares
different prices and dispositions of the products on the shelf, but we might know that
for any combination of prices, a small change in just one price, results in a more
similar environment than a big change in all prices.
We show here that, even if the information regarding similarity comparisons is
partial, it is still possible to construct two sets that contain only conscious and automatic observations respectively, and that one consistency requirement characterizes
all DD processes. In order to do so, we assume that, if the individual follows a DD
process, the similarity σ cardinally represents a completion of such partial order.
Thus, for any e, e0 , g, g 0 ∈ E, (e, e0 )S(g, g 0 ) implies σ(e, e0 ) ≥ σ(g, g 0 ) and we say that
(e, e0 ) dominates (g, g 0 ). We first adapt the key concepts on which the algorithmic
analysis is based in order to encompass this new assumption.
The two concepts of familiarity need to be changed. In particular, given that it is
not always possible to define the most familiar past environment, the new familiarity
definitions will be sets containing undominated pairs of environments. Let F (t) and
31
F (t|at ) be defined as follows:
F (t) = {(et , es )|s < t, as ∈ At and there is no w < t such that (et , ew )S(et , es ) and aw ∈ At },
F (t|at ) = {(et , es )|s < t, as = at and there is no w < t such that (et , ew )S(et , es ) and aw = at }.
That is, F (t) and F (t|at ) generalize the idea behind f (t) and f (t|at ), respectively. In fact, F (t) contains all those undominated pairs of environments where
et is compared with past observations which choice could be replicated. Similarly,
F (t|at ) contains all those undominated pairs of environments where et is compared
with past observations which choice could have been replicated. We can easily redefine
the concept of link. We say that observation t is linked to the set of observations O
whenever either F (t|at ) = ∅ or for all (et , e) ∈ F (t|at ) there exists s ∈ O such that
(es , e0 )S(et , e), for some (es , e0 ) ∈ F (s). Two things are worth underlining. First, notice that F (t|at ) = ∅ only if t is new, thus, as in the main analysis, new observations
are linked with any other observation. Second notice that this time we defined the
link between an observation t and a set of observations O. This helps to understand
whether an observation is generated by the maximizing self once we know that another observation is. If all observations in O are generated consciously and for each
one of them there exists a pair of environments that dominates a pair in F (t|at ) then
it must be that the maximizing self generated t too. This is because for all observation s in O, the similarity of all pairs of environments contained in F (s) must be
below the similarity threshold.
Then, we say that observation t is Consciously-indirectly linked to the set of observations O if there exists a sequence of observations t1 , . . . , tk such that t = t1 , tk is
linked to O and ti is linked to {ti+1 , ti+2 , ..., tk }∪O for every i = 1, 2, . . . , k −1. Define
DN̂ as the set containing all new observations and all those observations indirectly
linked to the set of new observations. Proposition 2 shows that DN̂ contains only
observations generated by the maximizing self.
What about automatic choices? As in the main analysis, whenever a cycle is
present in the data, we know that at least one of the observations in the cycle must
be generated by the automatic self. This time, given that we assume only a partial
32
knowledge of the similarity comparisons, it is not always possible to define a most
familiar observation in a cycle.17 Nevertheless, notice that whenever an observation is
inconsistent with the revealed preference constructed from DN̂ , it must be that such
observation is automatically generated. Thus, say that observation t is cloned if it is
either a most familiar in a cycle or xR(t)y while yR(DN̂ )x.
Say that observation t is Automatically-indirectly linked to observation s if there
exists a sequence of observations t1 , . . . , tk such that t = t1 , tk = s and ti is linked to
ti+1 for every i = 1, 2, . . . , k−1. Whenever we know that observation t is automatically
generated, we can infer that observation s is generated by the automatic self too, only
if for all pairs of environments in F (t|At ) there exists a pair of environments in F (s)
that dominates it. In fact, in general, only the similarity of some pairs of environments
contained in F (t|at ) is above the similarity threshold. As before, let DĈ be the set
containing all cloned observations and the observations to which they are indirectly
linked. Proposition 2 below shows that DĈ contains only observations generated by
the automatic self.
Proposition 2 For every collection of observations D generated by a DD process
where only a partial preorder over pairs of environments is known:
1. all decisions in DN̂ are generated by the maximizing self and all decisions in
DĈ are generated by the automatic self,
2. if x is revealed preferred to y for the set of observations DN̂ , then x y.
Proof. First we show that any observation t linked to a set O of conscious observations must be generated by the maximizing self too. In fact, notice that for any s ∈ O
we know that σ(es , e0 ) ≤ α for all (es , e0 ) ∈ F (s). Then, given t is linked to O we know
that for any (et , e) ∈ F (t|at ) there exists s ∈ O such that (es , e0 )S(et , e), for some
(es , e0 ) ∈ F (s). Now, given the definition of F (t|at ) this implies that σ(et , ew ) ≤ α
for all w < t such that aw = at and the result follows. Then, by Proposition 1 in
the paper we know that new observations are consciously generated and applying the
17
Obviously, in this context, a most familiar observation in a cycle would be an observation t
belonging to a cycle such that for any other observation s in the cycle, F (t) dominates F (s). That
is, for any (es , e) ∈ F (s) there exists (et , e0 ) ∈ F (t) such that (et , e0 )S(es , e).
33
previous reasoning iteratively it is shown that DN̂ must contain only observations
generated by the maximizing self.
As a second step, we show that any observation s to which an observation t
generated by the automatic self is linked, must be automatically generated too. Given
t is generated by the automatic self it means that there exists a w < t such that
σ(et , ew ) > α and aw = at . Then, either (et , ew ) ∈ F (t|at ) or (et , ew ) ∈
/ F (t|at ).
• Let (et , ew ) ∈ F (t|at ). Then, given t is linked to s, there exists a pair (es , e0 ) ∈
F (s) such that (es , e0 )S(et , ew ). This implies σ(es , e0 ) ≥ σ(et , ew ) > α, and the
result follows.
• Let (et , ew ) ∈
/ F (t|at ). Then, there exists a w0 < t such that (et , ew0 )S(et , ew )
and (et , ew0 ) ∈ F (t|at ). This implies that σ(et , ew0 ) > σ(et , ew ) > α. Then, given
t is linked to s we know that for all (et , e) ∈ F (t|at ), there exists a (es , e0 ) ∈ F (s)
such that (es , e0 )S(et , e). In particular, there exists a (es , e0 ) ∈ F (s) such that
(es , e0 )S(et , ew0 ). This implies σ(es , e0 ) ≥ σ(et , ew0 ) > α, and the result follows.
Then, given that cloned observations are automatically generated, we can apply the
previous reasoning iteratively to show that DĈ must contain only observations generated by the automatic self.
Finally, by a reasoning similar to the one developed in the proof of Proposition 1
in the main text, given all observations in DN̂ must be consciously generated, R(DN̂ )
reveals the preference of the DM.
Thus, we see that knowing only a partial preorder does not heavily affect the structure of the algorithm and the main logical steps behind it. What is of interest is that
even with this assumption it is possible to characterize a DD process with just one
single condition, that is DN̂ -Consistency.
Axiom 2 (DN̂ -Consistency) A sequence of observations {(At , et , at )}Tt=1 satisfies
DN̂ -Consistency whenever xR(DN̂ )y implies not yR(DN̂ )x.
DN̂ -Consistency imposes asymmetry on the revealed preference obtained from DN̂ . If
a sequence of decision problems satisfies DN̂ -Consistency when only a partial preorder
34
is known, then we are able to characterize the preferences of the individual, the
similarity threshold and, more importantly, a similarity function that respects such
preorder. This is what the next theorem states. Notice that S is assumed to be
known.
Theorem 2 A sequence of observations {(At , et , at )}Tt=1 satisfies DN̂ -Consistency if
and only if there exist a preference relation , a similarity function σ representing S
and a similarity threshold α that characterize a DD process.
Proof. Necessity: Suppose that the sequence {(At , et , at )}Tt=1 is generated by a DD
process. Then it satisfies DN̂ -Consistency given that, according to Proposition 2, DN̂
contains only conscious observations and is a linear order.
Sufficiency: Suppose that the sequence {(At , et , at )}Tt=1 satisfies DN̂ -Consistency.
We need to show that it can be explained as if generated by a DD process. Notice
that DN̂ -Consistency implies that the revealed preference relation defined over DN̂ ,
i.e., R(DN̂ ), is asymmetric. Thus, by standard mathematical results, we can find a
transitive completion of R(DN̂ ), call it . By construction, all decisions in DN̂ can
be seen as the result of maximizing over the corresponding menu.
We now define σ. We first complete S. Notice that by construction of DN̂ , for
all t ∈
/ DN̂ there exists a pair (et , e) ∈ F (t|at ) such that there is no s ∈ DN̂ for which
(es , e0 )S(et , e), for some (es , e0 ) ∈ F (s). That is, for all observations not in DN̂ there
exists a pair of environments that is not dominated by any pair of environments of
observations in DN̂ , a pair that we call undominated. Then, let S 0 be the following
reflexive binary relation. For any undominated pair (et , e) ∈ F (t|at ) with t ∈
/ DN̂ ,
let for all s ∈ DN̂ and for all (es , e0 ) ∈ F (s), (et , e)S 0 (es , e0 ) and not (es , e0 )S 0 (et , e).
Let S 00 be the transitive closure of S ∪ S 0 . Notice that S 00 is an extension of S that
preserves its reflexivity and transitivity. Thus we can find a completion S ∗ of S 00 and
a similarity function σ : E × E → [0, 1] that represents S ∗ .
Finally, we can define α. For any observation t, let f ∗ (t) be as follows:
f ∗ (t) =
max σ(et , es ),
s<t,as ∈At
Then let α = maxt∈DN̂ f ∗ (t). Notice that by construction of σ for all t ∈
/ DN̂ there
35
exists a pair of environments (et , e) ∈ F (t|at ) such that for all s ∈ DN̂ , σ(et , e) >
f ∗ (s), hence σ(et , e) > α. So, for every observation not in DN̂ we can find a preceding
observation to imitate.
Thus, we can represent the choices as if generated by an individual with preference
relation , similarity function σ and similarity threshold α.
Intuitively, the observations in DN̂ are used to construct the preference relation of
the individual. The similarity function represents a possible extension of the partial
preorder that respects the absence of links between observations in DN̂ and the ones
outside that set. This is possible thanks to how DN̂ has been constructed and it
allows for the definition of the similarity threshold in a similar fashion as before.
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